library(forecast)
## Warning: package 'forecast' was built under R version 4.3.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(readxl)
library(TTR)

data <- read.csv('aggregated_data.csv')
ts_birth <- ts(data$Records,start=c(1985,1),frequency=12)
plot(ts_birth, main="Birth time series", ylab = "Birth data")

data
##     MonthYear Records
## 1      1-1985  300955
## 2      2-1985  278130
## 3      3-1985  307665
## 4      4-1985  298732
## 5      5-1985  314496
## 6      6-1985  305428
## 7      7-1985  330560
## 8      8-1985  333851
## 9      9-1985  328419
## 10    10-1985  320085
## 11    11-1985  298341
## 12    12-1985  307827
## 13     1-1986  303023
## 14     2-1986  279094
## 15     3-1986  309253
## 16     4-1986  300861
## 17     5-1986  312877
## 18     6-1986  304425
## 19     7-1986  331646
## 20     8-1986  331023
## 21     9-1986  331313
## 22    10-1986  315979
## 23    11-1986  289749
## 24    12-1986  309551
## 25     1-1987  301771
## 26     2-1987  280001
## 27     3-1987  314156
## 28     4-1987  304190
## 29     5-1987  316477
## 30     6-1987  317857
## 31     7-1987  333279
## 32     8-1987  328299
## 33     9-1987  331017
## 34    10-1987  323163
## 35    11-1987  302905
## 36    12-1987  317024
## 37     1-1988  306881
## 38     2-1988  294790
## 39     3-1988  318630
## 40     4-1988  305777
## 41     5-1988  322442
## 42     6-1988  325613
## 43     7-1988  343503
## 44     8-1988  351203
## 45     9-1988  343779
## 46    10-1988  328108
## 47    11-1988  311044
## 48    12-1988  318654
## 49     1-1989  320422
## 50     2-1989  300391
## 51     3-1989  339912
## 52     4-1989  318779
## 53     5-1989  336320
## 54     6-1989  338973
## 55     7-1989  356716
## 56     8-1989  366579
## 57     9-1989  357344
## 58    10-1989  344161
## 59    11-1989  325543
## 60    12-1989  335818
## 61     1-1990  335274
## 62     2-1990  312611
## 63     3-1990  350614
## 64     4-1990  336382
## 65     5-1990  354114
## 66     6-1990  347355
## 67     7-1990  367670
## 68     8-1990  372516
## 69     9-1990  358682
## 70    10-1990  353166
## 71    11-1990  333146
## 72    12-1990  336682
## 73     1-1991  335172
## 74     2-1991  309130
## 75     3-1991  344079
## 76     4-1991  335626
## 77     5-1991  353131
## 78     6-1991  334265
## 79     7-1991  362913
## 80     8-1991  366786
## 81     9-1991  356016
## 82    10-1991  348934
## 83    11-1991  323635
## 84    12-1991  341220
## 85     1-1992  334045
## 86     2-1992  315448
## 87     3-1992  339518
## 88     4-1992  333373
## 89     5-1992  344137
## 90     6-1992  339664
## 91     7-1992  359112
## 92     8-1992  348949
## 93     9-1992  347547
## 94    10-1992  343546
## 95    11-1992  321943
## 96    12-1992  337732
## 97     1-1993  323073
## 98     2-1993  304656
## 99     3-1993  342187
## 100    4-1993  327042
## 101    5-1993  335989
## 102    6-1993  335349
## 103    7-1993  352554
## 104    8-1993  350898
## 105    9-1993  348013
## 106   10-1993  332937
## 107   11-1993  316379
## 108   12-1993  331163
## 109    1-1994  320700
## 110    2-1994  301325
## 111    3-1994  339734
## 112    4-1994  317387
## 113    5-1994  330294
## 114    6-1994  329736
## 115    7-1994  345859
## 116    8-1994  352171
## 117    9-1994  339219
## 118   10-1994  330168
## 119   11-1994  319395
## 120   12-1994  326743
## 121    1-1995  316011
## 122    2-1995  295094
## 123    3-1995  328502
## 124    4-1995  309117
## 125    5-1995  334541
## 126    6-1995  329800
## 127    7-1995  340872
## 128    8-1995  350734
## 129    9-1995  339100
## 130   10-1995  330011
## 131   11-1995  310815
## 132   12-1995  314969
## 133    1-1996  314283
## 134    2-1996  301763
## 135    3-1996  322581
## 136    4-1996  312595
## 137    5-1996  325708
## 138    6-1996  318525
## 139    7-1996  345162
## 140    8-1996  346317
## 141    9-1996  336348
## 142   10-1996  336346
## 143   11-1996  309397
## 144   12-1996  322469
## 145    1-1997  317211
## 146    2-1997  291541
## 147    3-1997  321212
## 148    4-1997  314230
## 149    5-1997  330331
## 150    6-1997  321867
## 151    7-1997  346506
## 152    8-1997  339122
## 153    9-1997  333600
## 154   10-1997  328657
## 155   11-1997  307282
## 156   12-1997  329335
## 157    1-1998  319340
## 158    2-1998  298711
## 159    3-1998  329436
## 160    4-1998  319758
## 161    5-1998  330519
## 162    6-1998  327091
## 163    7-1998  348651
## 164    8-1998  344736
## 165    9-1998  343384
## 166   10-1998  332790
## 167   11-1998  313241
## 168   12-1998  333896
## 169    1-1999  319182
## 170    2-1999  297568
## 171    3-1999  332939
## 172    4-1999  316889
## 173    5-1999  328526
## 174    6-1999  332201
## 175    7-1999  349812
## 176    8-1999  351371
## 177    9-1999  349409
## 178   10-1999  332980
## 179   11-1999  315289
## 180   12-1999  333251
## 181    1-2000  330108
## 182    2-2000  317377
## 183    3-2000  340553
## 184    4-2000  317180
## 185    5-2000  341207
## 186    6-2000  341206
## 187    7-2000  348975
## 188    8-2000  360080
## 189    9-2000  347609
## 190   10-2000  343921
## 191   11-2000  333811
## 192   12-2000  336787
## 193    1-2001  335198
## 194    2-2001  303534
## 195    3-2001  338684
## 196    4-2001  323613
## 197    5-2001  344017
## 198    6-2001  331085
## 199    7-2001  351047
## 200    8-2001  361802
## 201    9-2001  342564
## 202   10-2001  344074
## 203   11-2001  323746
## 204   12-2001  326569
## 205    1-2002  330674
## 206    2-2002  303977
## 207    3-2002  331505
## 208    4-2002  324432
## 209    5-2002  339007
## 210    6-2002  327588
## 211    7-2002  357669
## 212    8-2002  359417
## 213    9-2002  348814
## 214   10-2002  345814
## 215   11-2002  318573
## 216   12-2002  334256
## 217    1-2003  315731
## 218    2-2003  294026
## 219    3-2003  322033
## 220    4-2003  315525
## 221    5-2003  331114
## 222    6-2003  322097
## 223    7-2003  347943
## 224    8-2003  344573
## 225    9-2003  344243
## 226   10-2003  338733
## 227   11-2003  306617
## 228   12-2003  329151
## 229    1-2004  318661
## 230    2-2004  301853
## 231    3-2004  330772
## 232    4-2004  318200
## 233    5-2004  322256
## 234    6-2004  329332
## 235    7-2004  343675
## 236    8-2004  340150
## 237    9-2004  340857
## 238   10-2004  333713
## 239   11-2004  321667
## 240   12-2004  331336
## 241    1-2005  317318
## 242    2-2005  295998
## 243    3-2005  333565
## 244    4-2005  317561
## 245    5-2005  330399
## 246    6-2005  335206
## 247    7-2005  341355
## 248    8-2005  353228
## 249    9-2005  347697
## 250   10-2005  329836
## 251   11-2005  321105
## 252   12-2005  333319
## 253    1-2006  325698
## 254    2-2006  305472
## 255    3-2006  341008
## 256    4-2006  314750
## 257    5-2006  338949
## 258    6-2006  341884
## 259    7-2006  351612
## 260    8-2006  370881
## 261    9-2006  358503
## 262   10-2006  351308
## 263   11-2006  336795
## 264   12-2006  340876
## 265    1-2007  339498
## 266    2-2007  312544
## 267    3-2007  344586
## 268    4-2007  322728
## 269    5-2007  345489
## 270    6-2007  342184
## 271    7-2007  362651
## 272    8-2007  373161
## 273    9-2007  350856
## 274   10-2007  353187
## 275   11-2007  338699
## 276   12-2007  339001
## 277    1-2008  340881
## 278    2-2008  323131
## 279    3-2008  334464
## 280    4-2008  330277
## 281    5-2008  338343
## 282    6-2008  332382
## 283    7-2008  358201
## 284    8-2008  356496
## 285    9-2008  351705
## 286   10-2008  341755
## 287   11-2008  309489
## 288   12-2008  338449
## 289    1-2009  322970
## 290    2-2009  302399
## 291    3-2009  331405
## 292    4-2009  321409
## 293    5-2009  328731
## 294    6-2009  330637
## 295    7-2009  351254
## 296    8-2009  343063
## 297    9-2009  345608
## 298   10-2009  331806
## 299   11-2009  305917
## 300   12-2009  326161
## 301    1-2010  308846
## 302    2-2010  288355
## 303    3-2010  322930
## 304    4-2010  309541
## 305    5-2010  312567
## 306    6-2010  318820
## 307    7-2010  329198
## 308    8-2010  333829
## 309    9-2010  334714
## 310   10-2010  321907
## 311   11-2010  311773
## 312   12-2010  323854
## 313    1-2011  306090
## 314    2-2011  284436
## 315    3-2011  314715
## 316    4-2011  298431
## 317    5-2011  311028
## 318    6-2011  321093
## 319    7-2011  329242
## 320    8-2011  342850
## 321    9-2011  329598
## 322   10-2011  313550
## 323   11-2011  307088
## 324   12-2011  312567
## 325    1-2012  301951
## 326    2-2012  290568
## 327    3-2012  308944
## 328    4-2012  292182
## 329    5-2012  314235
## 330    6-2012  311706
## 331    7-2012  331255
## 332    8-2012  344369
## 333    9-2012  324607
## 334   10-2012  329634
## 335   11-2012  310492
## 336   12-2012  309741
## 337    1-2013  308994
## 338    2-2013  278307
## 339    3-2013  305462
## 340    4-2013  297011
## 341    5-2013  313635
## 342    6-2013  304497
## 343    7-2013  332739
## 344    8-2013  337261
## 345    9-2013  322855
## 346   10-2013  324983
## 347   11-2013  304216
## 348   12-2013  320526
## 349    1-2014  311524
## 350    2-2014  284520
## 351    3-2014  308546
## 352    4-2014  303969
## 353    5-2014  318680
## 354    6-2014  310606
## 355    7-2014  339057
## 356    8-2014  337677
## 357    9-2014  332602
## 358   10-2014  327849
## 359   11-2014  303948
## 360   12-2014  325355
## 361    1-2015  311182
## 362    2-2015  284470
## 363    3-2015  313665
## 364    4-2015  305815
## 365    5-2015  312181
## 366    6-2015  314757
## 367    7-2015  336869
## 368    8-2015  335885
## 369    9-2015  331542
## 370   10-2015  323570
## 371   11-2015  304502
## 372   12-2015  320458
Acf(ts_birth, lag.max=48, main = "Birth data ACF")

fivenum(ts_birth)
## [1] 278130.0 314605.5 329517.0 339823.0 373161.0
mean(ts_birth)
## [1] 327413.5
birth_mean <- mean(ts_birth)  
birth_median <- median(ts_birth)


boxplot(ts_birth, main="US birth data box plot",ylab="Birth records")

hist(ts_birth, main="Birth data histogram",xlab="Birth records")
abline(v = birth_mean, col = "red", lwd = 1, lty = 1)   
abline(v = birth_median, col = "blue", lwd = 1, lty = 1)

decomp <- decompose(ts_birth)
plot(decomp)

attributes(decomp)
## $names
## [1] "x"        "seasonal" "trend"    "random"   "figure"   "type"    
## 
## $class
## [1] "decomposed.ts"
decomp$trend
##           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
## 1985       NA       NA       NA       NA       NA       NA 310460.3 310586.6
## 1986 310763.2 310690.7 310693.4 310642.9 310113.8 309827.7 309847.3 309833.0
## 1987 312044.1 311998.7 311872.8 312159.8 313007.3 313866.9 314391.2 315220.3
## 1988 317911.0 319291.3 320777.4 321515.2 322060.4 322467.4 323099.5 323897.1
## 1989 329807.9 330999.1 332205.0 333439.0 334712.0 336031.3 337365.3 338493.3
## 1990 343999.0 344702.8 345005.9 345436.9 346128.9 346481.7 346513.4 346364.1
## 1991 344240.5 343803.6 343453.8 343166.3 342593.7 342386.5 342528.6 342744.9
## 1992 341982.4 341080.8 339984.7 339407.3 339112.3 338896.5 338294.0 337387.2
## 1993 335320.5 335128.5 335229.1 334806.5 334132.6 333627.0 333254.5 333016.8
## 1994 330647.7 330421.8 330108.4 329626.6 329636.9 329578.4 329198.9 328743.9
## 1995 327010.5 326742.9 326678.0 326666.5 326302.5 325454.4 324891.8 325097.7
## 1996 323675.1 323669.8 323371.1 323520.4 323725.3 323978.7 324413.2 324109.2
## 1997 324425.2 324181.5 323767.2 323332.3 322923.8 323121.8 323496.5 323884.0
## 1998 325869.1 326192.4 326834.0 327413.9 327834.4 328272.7 328456.2 328402.0
## 1999 328715.3 329040.1 329567.6 329826.6 329919.8 329978.3 330406.7 331687.3
## 2000 334943.7 335271.7 335559.6 335940.5 337168.1 338087.2 338446.6 338081.9
## 2001 337362.5 337520.6 337382.1 337178.3 336765.3 335920.2 335305.9 335135.9
## 2002 334191.3 334367.9 334528.9 334861.8 334718.8 334823.5 334521.2 333484.0
## 2003 330017.2 328993.4 328184.5 327699.0 326905.8 326194.9 326104.2 326552.5
## 2004 327516.7 327154.5 326829.2 326478.9 326896.8 327615.0 327650.0 327350.1
## 2005 328357.1 328805.3 329635.2 329758.7 329573.8 329633.0 330064.7 330808.7
## 2006 333285.8 334448.7 335634.5 336979.4 338527.8 339496.5 340386.3 341256.0
## 2007 343543.6 344098.6 343875.0 343634.6 343792.2 343793.5 343773.0 344271.7
## 2008 342900.7 342020.9 341361.9 340920.9 339227.5 337987.4 337218.1 335608.0
## 2009 332514.4 331665.2 330851.5 330182.9 329619.5 328958.7 327858.2 326684.5
## 2010 321153.3 319849.6 319010.9 318144.5 317976.1 318124.0 317913.0 317634.9
## 2011 315924.2 316301.9 316464.6 315903.2 315359.8 314694.3 314051.5 314134.6
## 2012 312957.3 313104.5 312959.8 313422.0 314234.0 314258.1 314433.8 314216.4
## 2013 313228.8 312994.5 312625.3 312358.5 311903.2 312091.1 312645.9 313010.2
## 2014 315298.7 315579.2 316002.7 316528.2 316636.5 316826.5 317013.5 316997.2
## 2015 317288.7 317122.8 317004.0 316781.5 316626.3 316445.4       NA       NA
##           Sep      Oct      Nov      Dec
## 1985 310692.9 310847.8 310869.0 310759.8
## 1986 310075.0 310418.0 310706.8 311416.4
## 1987 316022.9 316275.5 316590.1 317161.8
## 1988 325017.2 326445.8 327565.8 328700.7
## 1989 339448.4 340627.8 342102.7 343193.3
## 1990 345946.8 345643.0 345570.5 344984.2
## 1991 342818.1 342534.2 342065.6 341915.8
## 1992 337048.7 336896.1 336292.8 335773.5
## 1993 332775.8 332271.3 331631.7 331160.5
## 1994 328016.2 327203.7 327036.0 327215.7
## 1995 325128.9 325027.1 324804.0 323966.1
## 1996 323626.3 323637.4 323898.1 324230.0
## 1997 324525.4 325098.4 325336.6 325562.1
## 1998 328500.3 328526.7 328324.1 328454.0
## 1999 332829.9 333159.3 333699.8 334603.4
## 2000 337427.2 337617.4 338002.5 337697.9
## 2001 334855.2 334590.2 334415.6 334061.1
## 2002 332674.7 331908.9 331208.9 330651.2
## 2003 327242.7 327718.3 327460.7 327393.0
## 2004 327222.5 327312.3 327625.0 328209.0
## 2005 331513.5 331706.5 331945.7 332580.2
## 2006 341699.8 342181.2 342786.2 343071.2
## 2007 344291.1 344183.9 344200.7 343494.5
## 2008 334616.7 334119.8 333349.8 332876.5
## 2009 325746.2 324898.6 323730.6 322564.7
## 2010 317129.3 316324.1 315797.0 315827.6
## 2011 314149.6 313648.8 313522.0 313264.5
## 2012 313560.4 313616.5 313792.8 313467.4
## 2013 313397.6 313816.0 314316.1 314780.9
## 2014 317208.4 317498.6 317304.7 317206.9
## 2015       NA       NA       NA       NA
decomp$seasonal
##              Jan         Feb         Mar         Apr         May         Jun
## 1985  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1986  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1987  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1988  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1989  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1990  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1991  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1992  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1993  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1994  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1995  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1996  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1997  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1998  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 1999  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2000  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2001  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2002  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2003  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2004  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2005  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2006  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2007  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2008  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2009  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2010  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2011  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2012  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2013  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2014  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
## 2015  -8176.5471 -29475.5346    600.6362 -12440.5416   1966.5279   -707.6721
##              Jul         Aug         Sep         Oct         Nov         Dec
## 1985  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1986  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1987  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1988  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1989  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1990  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1991  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1992  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1993  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1994  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1995  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1996  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1997  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1998  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 1999  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2000  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2001  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2002  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2003  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2004  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2005  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2006  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2007  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2008  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2009  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2010  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2011  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2012  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2013  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2014  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
## 2015  18552.3890  21952.8459  14356.7071   6574.5029 -12800.8193   -402.4943
decomp$type
## [1] "additive"
s_adj <- seasadj(decomp)
plot(ts_birth,
      main = 'Seasonally adjusted time series',
      xlab = 'Time',
      ylab = 'Birth records')
lines(s_adj, col='red')

naive_f <- naive(ts_birth,60)

plot(naive_f$residuals, xlab="Time", ylab="Residuals")

hist(naive_f$residuals, xlab="Residuals", main="Residuals histogram")

naive_f$fitted
##         Jan    Feb    Mar    Apr    May    Jun    Jul    Aug    Sep    Oct
## 1985     NA 300955 278130 307665 298732 314496 305428 330560 333851 328419
## 1986 307827 303023 279094 309253 300861 312877 304425 331646 331023 331313
## 1987 309551 301771 280001 314156 304190 316477 317857 333279 328299 331017
## 1988 317024 306881 294790 318630 305777 322442 325613 343503 351203 343779
## 1989 318654 320422 300391 339912 318779 336320 338973 356716 366579 357344
## 1990 335818 335274 312611 350614 336382 354114 347355 367670 372516 358682
## 1991 336682 335172 309130 344079 335626 353131 334265 362913 366786 356016
## 1992 341220 334045 315448 339518 333373 344137 339664 359112 348949 347547
## 1993 337732 323073 304656 342187 327042 335989 335349 352554 350898 348013
## 1994 331163 320700 301325 339734 317387 330294 329736 345859 352171 339219
## 1995 326743 316011 295094 328502 309117 334541 329800 340872 350734 339100
## 1996 314969 314283 301763 322581 312595 325708 318525 345162 346317 336348
## 1997 322469 317211 291541 321212 314230 330331 321867 346506 339122 333600
## 1998 329335 319340 298711 329436 319758 330519 327091 348651 344736 343384
## 1999 333896 319182 297568 332939 316889 328526 332201 349812 351371 349409
## 2000 333251 330108 317377 340553 317180 341207 341206 348975 360080 347609
## 2001 336787 335198 303534 338684 323613 344017 331085 351047 361802 342564
## 2002 326569 330674 303977 331505 324432 339007 327588 357669 359417 348814
## 2003 334256 315731 294026 322033 315525 331114 322097 347943 344573 344243
## 2004 329151 318661 301853 330772 318200 322256 329332 343675 340150 340857
## 2005 331336 317318 295998 333565 317561 330399 335206 341355 353228 347697
## 2006 333319 325698 305472 341008 314750 338949 341884 351612 370881 358503
## 2007 340876 339498 312544 344586 322728 345489 342184 362651 373161 350856
## 2008 339001 340881 323131 334464 330277 338343 332382 358201 356496 351705
## 2009 338449 322970 302399 331405 321409 328731 330637 351254 343063 345608
## 2010 326161 308846 288355 322930 309541 312567 318820 329198 333829 334714
## 2011 323854 306090 284436 314715 298431 311028 321093 329242 342850 329598
## 2012 312567 301951 290568 308944 292182 314235 311706 331255 344369 324607
## 2013 309741 308994 278307 305462 297011 313635 304497 332739 337261 322855
## 2014 320526 311524 284520 308546 303969 318680 310606 339057 337677 332602
## 2015 325355 311182 284470 313665 305815 312181 314757 336869 335885 331542
##         Nov    Dec
## 1985 320085 298341
## 1986 315979 289749
## 1987 323163 302905
## 1988 328108 311044
## 1989 344161 325543
## 1990 353166 333146
## 1991 348934 323635
## 1992 343546 321943
## 1993 332937 316379
## 1994 330168 319395
## 1995 330011 310815
## 1996 336346 309397
## 1997 328657 307282
## 1998 332790 313241
## 1999 332980 315289
## 2000 343921 333811
## 2001 344074 323746
## 2002 345814 318573
## 2003 338733 306617
## 2004 333713 321667
## 2005 329836 321105
## 2006 351308 336795
## 2007 353187 338699
## 2008 341755 309489
## 2009 331806 305917
## 2010 321907 311773
## 2011 313550 307088
## 2012 329634 310492
## 2013 324983 304216
## 2014 327849 303948
## 2015 323570 304502
naive_f$residuals
##         Jan    Feb    Mar    Apr    May    Jun    Jul    Aug    Sep    Oct
## 1985     NA -22825  29535  -8933  15764  -9068  25132   3291  -5432  -8334
## 1986  -4804 -23929  30159  -8392  12016  -8452  27221   -623    290 -15334
## 1987  -7780 -21770  34155  -9966  12287   1380  15422  -4980   2718  -7854
## 1988 -10143 -12091  23840 -12853  16665   3171  17890   7700  -7424 -15671
## 1989   1768 -20031  39521 -21133  17541   2653  17743   9863  -9235 -13183
## 1990   -544 -22663  38003 -14232  17732  -6759  20315   4846 -13834  -5516
## 1991  -1510 -26042  34949  -8453  17505 -18866  28648   3873 -10770  -7082
## 1992  -7175 -18597  24070  -6145  10764  -4473  19448 -10163  -1402  -4001
## 1993 -14659 -18417  37531 -15145   8947   -640  17205  -1656  -2885 -15076
## 1994 -10463 -19375  38409 -22347  12907   -558  16123   6312 -12952  -9051
## 1995 -10732 -20917  33408 -19385  25424  -4741  11072   9862 -11634  -9089
## 1996   -686 -12520  20818  -9986  13113  -7183  26637   1155  -9969     -2
## 1997  -5258 -25670  29671  -6982  16101  -8464  24639  -7384  -5522  -4943
## 1998  -9995 -20629  30725  -9678  10761  -3428  21560  -3915  -1352 -10594
## 1999 -14714 -21614  35371 -16050  11637   3675  17611   1559  -1962 -16429
## 2000  -3143 -12731  23176 -23373  24027     -1   7769  11105 -12471  -3688
## 2001  -1589 -31664  35150 -15071  20404 -12932  19962  10755 -19238   1510
## 2002   4105 -26697  27528  -7073  14575 -11419  30081   1748 -10603  -3000
## 2003 -18525 -21705  28007  -6508  15589  -9017  25846  -3370   -330  -5510
## 2004 -10490 -16808  28919 -12572   4056   7076  14343  -3525    707  -7144
## 2005 -14018 -21320  37567 -16004  12838   4807   6149  11873  -5531 -17861
## 2006  -7621 -20226  35536 -26258  24199   2935   9728  19269 -12378  -7195
## 2007  -1378 -26954  32042 -21858  22761  -3305  20467  10510 -22305   2331
## 2008   1880 -17750  11333  -4187   8066  -5961  25819  -1705  -4791  -9950
## 2009 -15479 -20571  29006  -9996   7322   1906  20617  -8191   2545 -13802
## 2010 -17315 -20491  34575 -13389   3026   6253  10378   4631    885 -12807
## 2011 -17764 -21654  30279 -16284  12597  10065   8149  13608 -13252 -16048
## 2012 -10616 -11383  18376 -16762  22053  -2529  19549  13114 -19762   5027
## 2013   -747 -30687  27155  -8451  16624  -9138  28242   4522 -14406   2128
## 2014  -9002 -27004  24026  -4577  14711  -8074  28451  -1380  -5075  -4753
## 2015 -14173 -26712  29195  -7850   6366   2576  22112   -984  -4343  -7972
##         Nov    Dec
## 1985 -21744   9486
## 1986 -26230  19802
## 1987 -20258  14119
## 1988 -17064   7610
## 1989 -18618  10275
## 1990 -20020   3536
## 1991 -25299  17585
## 1992 -21603  15789
## 1993 -16558  14784
## 1994 -10773   7348
## 1995 -19196   4154
## 1996 -26949  13072
## 1997 -21375  22053
## 1998 -19549  20655
## 1999 -17691  17962
## 2000 -10110   2976
## 2001 -20328   2823
## 2002 -27241  15683
## 2003 -32116  22534
## 2004 -12046   9669
## 2005  -8731  12214
## 2006 -14513   4081
## 2007 -14488    302
## 2008 -32266  28960
## 2009 -25889  20244
## 2010 -10134  12081
## 2011  -6462   5479
## 2012 -19142   -751
## 2013 -20767  16310
## 2014 -23901  21407
## 2015 -19068  15956
plot(naive_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(naive_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

Acf (naive_f$residuals, lag.max=48, main="Residuals ACF")

naive_ac <- accuracy(naive_f)
naive_forecast <- forecast(naive_f,h=12)
plot(ts_birth, main = "Birth time series",ylab="Birth records")
lines(naive_forecast$mean, col='red')

plot(naive_forecast)

# Moving average
plot(ts_birth, xlab="Time",ylab="Birth Records", main = "Time series and forecasts")

# I am using the simple moving average
# SMA: n =3, in red
sma_3 <- SMA(ts_birth,n= 3)
lines(sma_3, col='red')
sma3_f <- forecast(sma_3,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
# SMA: n =6, in blue
sma_6 <- SMA(ts_birth,n= 6)
lines(sma_6, col='blue')
sma6_f <- forecast(sma_6,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
# SMA: n =12, in green
sma_12 <- SMA(ts_birth,n= 12)
lines(sma_12, col='green')

sma12_f <- forecast(sma_12,h=60)
## Warning in ets(object, lambda = lambda, biasadj = biasadj,
## allow.multiplicative.trend = allow.multiplicative.trend, : Missing values
## encountered. Using longest contiguous portion of time series
plot(sma3_f)

plot(sma6_f)

plot(sma12_f)

plot(sma3_f$residuals, xlab="Time", ylab="Residuals")

hist(sma3_f$residuals, xlab="Residuals", main="Residuals histogram")

plot(sma3_f$residuals~sma3_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(sma3_f$residuals~sma3_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

Acf(sma3_f$residuals,lag.max=48,main="Residuals ACF")

sma3_f$residuals
##               Jan          Feb          Mar          Apr          May
## 1985                           -3242.962309    84.541714  1148.549027
## 1986  -830.224583  -896.984390   197.724160   602.666758   679.936901
## 1987   211.408905  2331.208574  1273.491521  2130.863055  1562.747451
## 1988  -540.102493  2834.428190   236.451990   936.798332 -1381.720849
## 1989  2377.320919  2000.556884  6805.520922   704.558159  1336.475276
## 1990  1952.310586  1226.057442  4646.789584  1641.684513  3200.602061
## 1991 -1048.241596 -2409.171639  2265.148674  1517.607744  4109.580790
## 1992   -20.414444  2850.760842  -835.874457  1118.975399 -1008.778113
## 1993 -1902.538737  -180.902374  1250.208048  2671.594740  -138.670752
## 1994   877.559313   562.664866  2613.320086   223.013999  -918.345423
## 1995   207.399730 -2537.047944   360.147387  -932.222855  2617.271798
## 1996  -272.886538  2586.894877  2290.870508   766.553180 -2597.894426
## 1997 -1470.541866  -386.073378  -665.037737   366.998796  2380.760676
## 1998  1792.882350  2667.019943  -277.650067  1442.861884    -3.557737
## 1999   362.319752   314.145251  -593.771710   571.536803  -251.568196
## 2000  3965.609253  6217.772672  2084.895554 -3062.026516 -2661.701512
## 2001  1967.330635 -4588.734600   380.450577 -2516.236102  2961.876181
## 2002   466.828139 -1022.338120  1406.671267  -734.829037  1124.778733
## 2003 -5109.799131 -2570.349841 -4260.731232  1377.883672  1870.569389
## 2004 -1770.891138  3990.345894   259.904038  1168.736808 -3788.701966
## 2005  -598.648748 -3034.341557   490.296678  1424.285989   892.310374
## 2006  3528.633651   302.662504  2267.643203 -2358.799001   580.491587
## 2007   929.978710 -2575.251521   966.335169 -4264.407139   449.834141
## 2008   792.306037   342.996038 -1794.357232 -2189.597466 -5467.597623
## 2009 -1294.428769  3248.781942 -2595.028942   859.967744 -1763.159257
## 2010 -2683.935000  -226.498735 -1274.304723  1637.734599 -2456.551379
## 2011  -394.186512 -3582.456755 -3287.543417 -1161.335602 -1644.594130
## 2012  1068.084833    54.116781 -1462.097908 -1895.082262 -2636.512149
## 2013 -2000.889766 -5180.225428 -1636.628340 -2594.846061  1288.327334
## 2014   365.689446 -1063.787737 -4284.496221 -1156.883746   853.745094
## 2015  -684.703699  -965.119139 -4167.423365  -413.416725 -1292.144850
##               Jun          Jul          Aug          Sep          Oct
## 1985  -574.429284   -57.326210  -336.577394  2452.311468   341.661492
## 1986 -1119.084727  -144.429790  -529.982244  3920.472157 -1267.865618
## 1987  1702.966732  -755.943640 -2669.178737  -659.321373   628.176188
## 1988  2824.691883  2134.949699  2956.889961   934.419308 -1209.013576
## 1989   122.669971  2183.794583  3438.959527   984.467896  -277.513831
## 1990  -664.717487   -36.721063  -493.803901 -1306.616732  -857.429782
## 1991 -2797.038551 -1307.343842 -2012.052026  2235.196881  -664.665407
## 1992   565.326788 -1823.703857 -4955.236099 -2354.548314 -1118.704417
## 1993 -1781.470211 -1890.053917 -1580.657546  -789.038147 -2508.216918
## 1994 -2819.693084  -880.720634   750.030453 -1867.074071 -1202.881340
## 1995   944.377729   172.743253 -1189.014282 -1943.381306   395.654427
## 1996  -824.990115   474.879778   305.638644   900.360273  1049.632151
## 1997   718.139151   339.175182 -3663.445163 -1113.034773 -1926.515254
## 1998  -302.826732  -794.427252 -1847.080420   393.886811 -1289.643862
## 1999   263.695885   567.914671  1030.200656   669.695575 -1639.588080
## 2000   725.333730   186.188603  -294.635598 -2918.904741  2334.148715
## 2001 -2025.304278 -1240.209510  -620.252740 -1193.211643  1702.654960
## 2002  -793.608608   684.680858   227.253219  2024.857477    16.962646
## 2003   573.680446   429.265993 -2073.126571  2367.985133   925.447543
## 2004    52.343690 -1894.520982  -578.111813 -1174.788016   709.463415
## 2005  1042.198504 -2494.958070  1040.198785  -890.223821   156.812464
## 2006   788.150960  1865.095990  4032.265391   421.180040  3828.914267
## 2007  -265.009824  2927.656880  2635.241276 -2194.532621   828.167195
## 2008  -102.853684 -1028.468284  -462.113746  1449.671306 -1461.668463
## 2009   293.459184  -425.544034 -1772.179831   -17.144825 -2459.211273
## 2010  -803.744917 -3796.339355   592.776806   310.698394  1605.996944
## 2011  2698.713142  -108.371323  4049.109019 -2238.756411 -1238.593049
## 2012  1489.338235  2656.048659  3465.775865  -783.040200  3428.307312
## 2013   236.597335  1544.909567  1311.195603  1070.852775  1391.534899
## 2014  1213.522882  1296.221169  -257.571924  2278.981222   225.206565
## 2015   915.803940   -24.889977  1344.101503   551.456418  -446.281667
##               Nov          Dec
## 1985  -383.665852 -2045.814792
## 1986 -2192.186035 -2326.531853
## 1987  3118.937701   222.610720
## 1988 -1846.642176 -3471.239866
## 1989 -2160.809540 -2285.872764
## 1990 -1543.048670 -2400.326735
## 1991 -2791.823978    23.000279
## 1992  2655.533822  1679.647846
## 1993   133.176299  -653.014814
## 1994   697.770987   786.275663
## 1995 -1708.925675 -3094.872415
## 1996  -738.099824   289.271688
## 1997  1014.299539  3523.620044
## 1998  1101.183969  1759.058287
## 1999  -445.972943  -462.715455
## 2000  2823.030545  1280.194741
## 2001 -1092.775845  -393.145651
## 2002 -2051.512127    72.024597
## 2003 -1076.664820  -107.043644
## 2004  5445.295688  1707.794762
## 2005   876.726242   119.304873
## 2006   131.639111 -1029.367167
## 2007    80.727584   955.117629
## 2008 -4047.926933   572.961928
## 2009  -747.847495 -1514.809701
## 2010  4250.269347  1270.391139
## 2011  -324.431073  -743.195444
## 2012   237.163190   -81.595182
## 2013   542.526432  4116.830036
## 2014   313.790517  2478.801160
## 2015  1123.178345  1213.260416
plot(sma6_f$residuals, xlab="Time", ylab="Residuals")

hist(sma6_f$residuals, xlab="Residuals", main="Residuals histogram")

plot(sma6_f$residuals~sma6_f$fitted, main = "Residual-Fitted values plot",xlab="Residuals",ylab="Fitted values")

plot(sma6_f$residuals~sma6_f$x, main = "Residual-Actual values plot",xlab="Residuals",ylab="Actual values")

attributes(sma6_f)
## $names
##  [1] "model"     "mean"      "level"     "x"         "upper"     "lower"    
##  [7] "fitted"    "method"    "series"    "residuals"
## 
## $class
## [1] "forecast"
sma6_f$residuals
##                Jan           Feb           Mar           Apr           May
## 1985                                                                      
## 1986 -1.063278e-03 -2.244091e-03 -3.080652e-03  1.414281e-04  7.638412e-05
## 1987 -1.648231e-03  4.387300e-04 -1.735079e-03  4.352518e-03  6.366863e-03
## 1988 -9.598788e-05  9.416404e-03 -5.399218e-04 -1.664365e-04  1.687900e-03
## 1989  1.324331e-03 -3.006865e-04  4.681111e-03  4.116022e-03  4.156820e-03
## 1990  1.550108e-03 -2.334662e-03  2.860855e-03  4.751801e-03  5.548347e-03
## 1991 -2.238602e-03 -5.148907e-03  8.137092e-04  1.712879e-03  3.231739e-03
## 1992 -6.560075e-04  4.551063e-04 -1.063113e-03  2.125580e-03  2.948980e-03
## 1993 -3.423232e-03  5.063436e-03  4.932321e-03  1.273457e-03 -5.209127e-04
## 1994 -1.421256e-03  2.127127e-03  3.601105e-03  2.045113e-03 -4.656590e-04
## 1995 -9.608935e-04 -2.273315e-03  2.424937e-03 -6.375836e-04  6.444170e-04
## 1996  1.323791e-03  2.236053e-03 -8.953621e-04  1.441667e-03  4.031640e-04
## 1997 -8.647642e-04 -1.859927e-03 -6.045595e-04 -1.478403e-03  3.579842e-03
## 1998 -8.150903e-05  5.915418e-03  4.472352e-03  4.325401e-03  3.058574e-03
## 1999 -1.563104e-03  2.220082e-03  1.481694e-03  1.247132e-03 -1.993416e-04
## 2000  3.295541e-03  8.312833e-03  1.046784e-03  2.698229e-04  4.455473e-03
## 2001  6.188441e-03 -3.406277e-03  2.088632e-03 -1.198039e-03 -2.621830e-03
## 2002  3.913309e-03 -3.021301e-03  1.919743e-03 -8.138241e-05  4.674980e-04
## 2003 -7.813056e-03 -6.467069e-03 -5.407922e-03 -4.138027e-03  8.307593e-04
## 2004 -1.342313e-03  4.620503e-03 -2.894167e-04 -1.145541e-03  2.860425e-04
## 2005 -9.687661e-05  3.521494e-03  2.741088e-03  6.431270e-04 -3.920339e-03
## 2006  5.495595e-03  1.188368e-03  2.897604e-03  1.059085e-03  6.264889e-04
## 2007  5.519146e-03 -5.317116e-03 -1.147321e-03 -5.558482e-03 -3.061074e-03
## 2008  1.965317e-03  9.673251e-05 -1.812949e-03 -2.176347e-03 -7.438525e-03
## 2009 -3.163367e-03 -6.654899e-05 -2.226466e-03  5.484429e-04  3.301567e-03
## 2010 -6.741358e-03  1.533232e-04 -3.111400e-03  1.928955e-05 -2.828711e-03
## 2011  1.701926e-03  7.253120e-04 -3.727507e-03 -2.325812e-03 -8.022411e-03
## 2012 -5.399620e-05 -2.410260e-04 -3.524410e-03 -6.947532e-04 -3.577207e-03
## 2013  1.339088e-03 -8.813272e-03 -2.770449e-03 -7.018310e-03 -5.182396e-03
## 2014  1.442218e-03 -2.130919e-03 -1.285591e-03 -1.800124e-03 -5.647173e-04
## 2015 -1.844774e-03 -1.830271e-03 -3.320509e-03 -1.731501e-03 -3.399290e-03
##                Jun           Jul           Aug           Sep           Oct
## 1985 -7.636175e-03 -1.464918e-04  1.296616e-03  2.636724e-03  2.995434e-04
## 1986 -1.259343e-03  8.262241e-04  4.384091e-04  4.404708e-03 -2.215692e-03
## 1987  4.072401e-03  8.433387e-04 -2.730333e-03  9.287137e-04 -4.403607e-04
## 1988  4.097935e-03  3.465657e-03  1.129320e-03  4.362635e-03  2.392921e-04
## 1989  9.327530e-03  1.982729e-03  5.030738e-03 -7.001992e-04  1.261908e-03
## 1990  4.342307e-03  3.394314e-04  2.332990e-03 -4.376044e-03 -1.766314e-03
## 1991 -6.244523e-04  2.717320e-04  3.154847e-03 -1.122249e-03 -2.533694e-03
## 1992 -6.341821e-04 -1.412228e-03 -8.552969e-03 -1.769183e-03 -2.937313e-03
## 1993 -8.662562e-04  9.668307e-04 -2.581623e-03 -3.652241e-03 -5.503879e-03
## 1994 -2.819385e-04 -1.195784e-03 -7.760286e-05 -6.897320e-03 -1.900973e-03
## 1995  2.340243e-03 -1.429208e-03  2.302188e-03 -1.730739e-03  1.247489e-03
## 1996  2.557224e-03  1.606917e-03 -3.699107e-03  1.876533e-04  2.755109e-03
## 1997 -2.537353e-05  7.522562e-04 -2.098811e-03 -6.307756e-04 -1.911343e-03
## 1998 -2.244594e-03 -1.590500e-04 -3.500606e-03 -2.462313e-04 -2.951774e-03
## 1999 -8.316522e-04  1.310713e-03  8.737563e-04  9.246101e-04 -1.634353e-03
## 2000  2.889776e-03 -5.964558e-03 -4.903384e-03 -3.304886e-03  4.305034e-03
## 2001 -2.440545e-03 -5.579466e-03  4.262071e-03 -4.548919e-03  1.672694e-03
## 2002  1.072605e-03 -3.273300e-04  2.025433e-03  1.494608e-03  1.015916e-03
## 2003 -4.318331e-03  4.260602e-03  6.704431e-04  5.059777e-03  2.446100e-03
## 2004  3.462999e-04 -1.527806e-03 -6.532433e-03 -1.100122e-03 -7.293680e-04
## 2005  2.202567e-03 -2.278792e-03  2.802065e-03 -2.636875e-04 -3.397323e-03
## 2006  3.906076e-03 -2.041873e-03  6.092994e-03  4.324503e-04  7.394927e-03
## 2007  1.381326e-03 -2.083185e-03  4.771789e-03 -3.846349e-03  5.845822e-03
## 2008 -1.989104e-03 -4.089243e-03 -7.416883e-03  3.647335e-03 -2.128463e-03
## 2009 -3.126421e-03  1.095929e-03 -4.801422e-03  1.186167e-03 -3.292423e-03
## 2010 -1.870683e-03 -2.259406e-03 -1.462990e-03  4.853700e-04 -1.914135e-03
## 2011  1.792659e-04 -1.372852e-03  4.704885e-03  8.799546e-04 -1.582901e-03
## 2012  9.039829e-04  1.594014e-03  1.667567e-03  1.107215e-03  9.752818e-03
## 2013 -8.091366e-04 -6.335056e-04  5.341525e-03  2.384933e-03  5.045975e-03
## 2014 -5.206901e-03  3.105899e-04  1.193779e-03  5.390512e-03  2.177574e-03
## 2015 -4.790190e-03 -6.722105e-05  8.385792e-04  2.705730e-03 -2.049719e-04
##                Nov           Dec
## 1985 -1.640006e-03  2.533668e-04
## 1986 -4.380826e-03  2.791122e-03
## 1987  2.114466e-04 -1.197858e-03
## 1988  3.470015e-04 -5.035166e-03
## 1989  1.459055e-04 -3.401915e-03
## 1990 -3.411631e-03 -5.408515e-03
## 1991 -6.653814e-03  4.027342e-03
## 1992 -2.215385e-03  1.356509e-04
## 1993 -1.086006e-03 -1.198482e-03
## 1994  3.069026e-03 -1.237176e-03
## 1995 -4.417523e-03 -7.247979e-03
## 1996 -8.481169e-04  1.629411e-03
## 1997 -3.713586e-03  4.218633e-03
## 1998 -1.001243e-03  3.331462e-03
## 1999  6.758714e-04 -6.393291e-06
## 2000  3.476996e-03 -2.971306e-03
## 2001 -2.112988e-03 -1.845497e-03
## 2002 -3.057396e-03  2.951309e-03
## 2003 -4.880679e-03  3.659144e-03
## 2004  7.985433e-03  4.420199e-04
## 2005  2.926576e-03 -1.456795e-03
## 2006  4.232220e-03 -2.834265e-03
## 2007  3.388884e-03 -2.580092e-03
## 2008 -5.378874e-03  4.474442e-03
## 2009 -2.884441e-03 -1.321397e-03
## 2010  8.650034e-03  2.408784e-03
## 2011  6.055698e-03 -5.087494e-03
## 2012  4.657916e-03 -2.796922e-03
## 2013  2.446209e-03  7.346906e-03
## 2014 -5.870696e-04  6.707683e-03
## 2015  3.966859e-03  2.344506e-03
plot(sma12_f$residuals, xlab="Time", ylab="Residuals")

hist(sma12_f$residuals, xlab="Residuals", main="Residuals histogram")

plot(sma12_f$residuals~sma12_f$fitted, main = "Residual-Fitted values plot",xlab="Residuals",ylab="Fitted values")

plot(sma12_f$residuals~sma12_f$x, main = "Residual-Actual values plot",xlab="Residuals",ylab="Actual values")

ses <- HoltWinters(ts_birth,beta = FALSE,gamma=FALSE)
ses_forecast <- forecast(ses,h=60)
ses$alpha
## [1] 0.5762466
ses_forecast$fitted[2]
## [1] 300955
sd(ses_forecast$residuals)
## [1] NA
plot(ses_forecast$residuals, xlab="Time", ylab="Residuals")

hist(ses_forecast$residuals, xlab="Residuals", main="Residuals histogram")

Acf (ses_forecast$residuals,lag.max=48,main="Residuals ACF")

plot(ses_forecast$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(ses_forecast$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

plot(ses_forecast, main ="Simple Exponential Smoothing forecast")

plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(ses_forecast$fitted,col='red')

HW <- HoltWinters(ts_birth)
HW_f <- forecast(HW,h=60)

HW$alpha
##     alpha 
## 0.2066015
HW$beta
##      beta 
## 0.1299084
HW$gamma
##     gamma 
## 0.1355967
plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(HW_f$fitted,col='red')

plot(HW_f$residuals, xlab="Time", ylab="Residuals")

hist(HW_f$residuals, xlab="Residuals", main="Residuals histogram")

Acf (HW_f$residuals, lag.max=48, main="Residuals ACF")

plot(HW_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(HW_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

plot(HW_f)

auto_fit <- auto.arima(ts_birth,trace=TRUE,stepwise=FALSE)
## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(0,1,0)(0,1,0)[12]                    : 7224.06
##  ARIMA(0,1,0)(0,1,1)[12]                    : Inf
##  ARIMA(0,1,0)(0,1,2)[12]                    : Inf
##  ARIMA(0,1,0)(1,1,0)[12]                    : 7231.767
##  ARIMA(0,1,0)(1,1,1)[12]                    : 7137.604
##  ARIMA(0,1,0)(1,1,2)[12]                    : Inf
##  ARIMA(0,1,0)(2,1,0)[12]                    : 7204.457
##  ARIMA(0,1,0)(2,1,1)[12]                    : 7040.941
##  ARIMA(0,1,0)(2,1,2)[12]                    : 6998.324
##  ARIMA(0,1,1)(0,1,0)[12]                    : 7067.526
##  ARIMA(0,1,1)(0,1,1)[12]                    : Inf
##  ARIMA(0,1,1)(0,1,2)[12]                    : Inf
##  ARIMA(0,1,1)(1,1,0)[12]                    : 7060.617
##  ARIMA(0,1,1)(1,1,1)[12]                    : 6970.324
##  ARIMA(0,1,1)(1,1,2)[12]                    : Inf
##  ARIMA(0,1,1)(2,1,0)[12]                    : 7050.7
##  ARIMA(0,1,1)(2,1,1)[12]                    : 6943.139
##  ARIMA(0,1,1)(2,1,2)[12]                    : 6934.887
##  ARIMA(0,1,2)(0,1,0)[12]                    : 7069.541
##  ARIMA(0,1,2)(0,1,1)[12]                    : 6953.398
##  ARIMA(0,1,2)(0,1,2)[12]                    : Inf
##  ARIMA(0,1,2)(1,1,0)[12]                    : 7056.93
##  ARIMA(0,1,2)(1,1,1)[12]                    : 6967.235
##  ARIMA(0,1,2)(1,1,2)[12]                    : 6956.061
##  ARIMA(0,1,2)(2,1,0)[12]                    : 7049.062
##  ARIMA(0,1,2)(2,1,1)[12]                    : 6943.594
##  ARIMA(0,1,3)(0,1,0)[12]                    : 7068.331
##  ARIMA(0,1,3)(0,1,1)[12]                    : 6955.453
##  ARIMA(0,1,3)(0,1,2)[12]                    : Inf
##  ARIMA(0,1,3)(1,1,0)[12]                    : 7057.802
##  ARIMA(0,1,3)(1,1,1)[12]                    : 6967.187
##  ARIMA(0,1,3)(2,1,0)[12]                    : 7049.636
##  ARIMA(0,1,4)(0,1,0)[12]                    : 7058.872
##  ARIMA(0,1,4)(0,1,1)[12]                    : 6937.713
##  ARIMA(0,1,4)(1,1,0)[12]                    : 7044.188
##  ARIMA(0,1,5)(0,1,0)[12]                    : 7040.125
##  ARIMA(1,1,0)(0,1,0)[12]                    : 7141.841
##  ARIMA(1,1,0)(0,1,1)[12]                    : Inf
##  ARIMA(1,1,0)(0,1,2)[12]                    : Inf
##  ARIMA(1,1,0)(1,1,0)[12]                    : 7134.709
##  ARIMA(1,1,0)(1,1,1)[12]                    : 7035.774
##  ARIMA(1,1,0)(1,1,2)[12]                    : Inf
##  ARIMA(1,1,0)(2,1,0)[12]                    : 7118.449
##  ARIMA(1,1,0)(2,1,1)[12]                    : 6997.593
##  ARIMA(1,1,0)(2,1,2)[12]                    : 6979.979
##  ARIMA(1,1,1)(0,1,0)[12]                    : 7070.422
##  ARIMA(1,1,1)(0,1,1)[12]                    : 6957.612
##  ARIMA(1,1,1)(0,1,2)[12]                    : Inf
##  ARIMA(1,1,1)(1,1,0)[12]                    : 7058.611
##  ARIMA(1,1,1)(1,1,1)[12]                    : 6966.577
##  ARIMA(1,1,1)(1,1,2)[12]                    : 6955.293
##  ARIMA(1,1,1)(2,1,0)[12]                    : 7050.613
##  ARIMA(1,1,1)(2,1,1)[12]                    : 6958.465
##  ARIMA(1,1,2)(0,1,0)[12]                    : 7045.333
##  ARIMA(1,1,2)(0,1,1)[12]                    : 6958.381
##  ARIMA(1,1,2)(0,1,2)[12]                    : Inf
##  ARIMA(1,1,2)(1,1,0)[12]                    : 7058.896
##  ARIMA(1,1,2)(1,1,1)[12]                    : 6966.876
##  ARIMA(1,1,2)(2,1,0)[12]                    : 7052.659
##  ARIMA(1,1,3)(0,1,0)[12]                    : 7043.631
##  ARIMA(1,1,3)(0,1,1)[12]                    : 6940.16
##  ARIMA(1,1,3)(1,1,0)[12]                    : 7029.352
##  ARIMA(1,1,4)(0,1,0)[12]                    : 7042.933
##  ARIMA(2,1,0)(0,1,0)[12]                    : 7078.83
##  ARIMA(2,1,0)(0,1,1)[12]                    : 6932.109
##  ARIMA(2,1,0)(0,1,2)[12]                    : Inf
##  ARIMA(2,1,0)(1,1,0)[12]                    : 7039.756
##  ARIMA(2,1,0)(1,1,1)[12]                    : 6949.61
##  ARIMA(2,1,0)(1,1,2)[12]                    : 6942.982
##  ARIMA(2,1,0)(2,1,0)[12]                    : 7030.294
##  ARIMA(2,1,0)(2,1,1)[12]                    : 6955.388
##  ARIMA(2,1,1)(0,1,0)[12]                    : 7071.88
##  ARIMA(2,1,1)(0,1,1)[12]                    : 6926.495
##  ARIMA(2,1,1)(0,1,2)[12]                    : Inf
##  ARIMA(2,1,1)(1,1,0)[12]                    : 7038.397
##  ARIMA(2,1,1)(1,1,1)[12]                    : 6946.29
##  ARIMA(2,1,1)(2,1,0)[12]                    : 7027.478
##  ARIMA(2,1,2)(0,1,0)[12]                    : 7043.649
##  ARIMA(2,1,2)(0,1,1)[12]                    : 6926.132
##  ARIMA(2,1,2)(1,1,0)[12]                    : 7029.284
##  ARIMA(2,1,3)(0,1,0)[12]                    : 7045.449
##  ARIMA(3,1,0)(0,1,0)[12]                    : 7081.8
##  ARIMA(3,1,0)(0,1,1)[12]                    : 6930.677
##  ARIMA(3,1,0)(0,1,2)[12]                    : Inf
##  ARIMA(3,1,0)(1,1,0)[12]                    : 7040.976
##  ARIMA(3,1,0)(1,1,1)[12]                    : 6952.285
##  ARIMA(3,1,0)(2,1,0)[12]                    : 7030.012
##  ARIMA(3,1,1)(0,1,0)[12]                    : 7067.075
##  ARIMA(3,1,1)(0,1,1)[12]                    : 6927.343
##  ARIMA(3,1,1)(1,1,0)[12]                    : 7040.14
##  ARIMA(3,1,2)(0,1,0)[12]                    : 7045.375
##  ARIMA(4,1,0)(0,1,0)[12]                    : 7049.838
##  ARIMA(4,1,0)(0,1,1)[12]                    : 6921.383
##  ARIMA(4,1,0)(1,1,0)[12]                    : 7027.863
##  ARIMA(4,1,1)(0,1,0)[12]                    : 7045.11
##  ARIMA(5,1,0)(0,1,0)[12]                    : 7045.571
## 
##  Now re-fitting the best model(s) without approximations...
## 
## 
## 
## 
##  Best model: ARIMA(4,1,0)(0,1,1)[12]
auto_f <- forecast(auto_fit,h=60,level=c(99.5)) 
plot(auto_f)

plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(auto_f$fitted,col='red')

plot(auto_f$residuals, xlab="Time", ylab="Residuals")

hist(auto_f$residuals, xlab="Residuals", main="Residuals histogram")

Acf (auto_f$residuals, lag.max = 48, main = "Residuals ACF")

plot(auto_f$residuals~naive_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(auto_f$residuals~naive_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

mean_f <- meanf(ts_birth,60)

plot(mean_f$residuals, xlab="Time", ylab="Residuals")

hist(mean_f$residuals, xlab="Residuals", main="Residuals histogram")

plot(mean_f$residuals~mean_f$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(mean_f$residuals~mean_f$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

Acf (mean_f$residuals, lag.max=48, main="Residuals ACF")

plot(mean_f)

stl_birth <- stl(ts_birth, s.window='periodic')
stl_forecast <- forecast(stl_birth, h=60)

plot(stl_forecast$residuals, xlab="Time", ylab="Residuals")

hist(stl_forecast$residuals, xlab="Residuals", main="Residuals histogram")

plot(stl_forecast$residuals~stl_forecast$fitted, main = "Residual-Fitted values plot",ylab="Residuals",xlab="Fitted values")

plot(stl_forecast$residuals~stl_forecast$x, main = "Residual-Actual values plot",ylab="Residuals",xlab="Actual values")

Acf (stl_forecast$residuals, lag.max=48,main="Residuals ACF")

plot(stl_forecast)

plot(ts_birth, main="Actual data & forecasts",ylab="Birth records")
lines(stl_forecast$fitted,col='red')

naive_ac <- accuracy(naive_f)
mean_ac <- accuracy(mean_f)
sma3_ac <- accuracy(sma_3,ts_birth)
sma6_ac <- accuracy(sma_6,ts_birth)
sma12_ac <- accuracy(sma_12,ts_birth)
ses_ac <- accuracy(ses$fitted,ts_birth)
HW_ac <- accuracy(HW$fitted,ts_birth)
stl_ac <- accuracy(stl_forecast$fitted,ts_birth)
arima_ac <- accuracy(auto_f)
naive_mape <- naive_ac['Training set', 'MAPE']
mean_mape <- mean_ac['Training set', 'MAPE']
sma3_mape <- sma3_ac['Test set', 'MAPE']
sma6_mape <- sma6_ac['Test set', 'MAPE']
sma12_mape <- sma12_ac['Test set', 'MAPE']
ses_mape <- ses_ac['Test set', 'MAPE']
HW_mape <- HW_ac['Test set', 'MAPE']
stl_mape <- stl_ac['Test set', 'MAPE']
arima_mape <- arima_ac['Training set','MAPE']
mape_table <- data.frame(
    Method = c("Naive","Mean", "SMA3", "SMA6", "SMA12", "SES", "HW","STL","ARIMA"),
    MAPE = c(naive_mape, mean_mape,sma3_mape, sma6_mape, sma12_mape, ses_mape, HW_mape,stl_mape,arima_mape),
    stringsAsFactors = FALSE
)

mape_table
##   Method     MAPE
## 1  Naive 4.303580
## 2   Mean 4.507706
## 3   SMA3 2.487378
## 4   SMA6 3.719305
## 5  SMA12 3.640532
## 6    SES 3.795787
## 7     HW 1.294030
## 8    STL 1.236984
## 9  ARIMA 1.167318